Adjoint theory & Solver in OpenFOAM GERMAN OPENFOAM USER MEETING 2017

CHRISTOS KAPELLOS christos.kapellos@volkswagen.de TU BRAUNSCHWEIG, 21.03.2017

2 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

AGENDA

• Adjoint theory: • Numerical Optimisation • Why do we need the adjoints? • Continuous adjoint formulation

• OpenFOAM Adjoint Solver:

• Our baseline: adjointShapeOptimizationFoam (version 2.3.1) • Enhance the code • Applications:

• pitzDaily • Airfoil 2D

3 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

INTRODUCTION

• Find the minimum of f(x)

• Analytical expression

• Here: 𝛻𝛻𝑓𝑓 = 2𝑥𝑥 , 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 = 0

NUMERICAL OPTIMISATION

4 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

INTRODUCTION

x0

𝜵𝜵𝒇𝒇

x1

f0

f1

𝜵𝜵𝒇𝒇

x2

f2

f3

x3

• In the computational world the analytic functions are not known

• A simulation is a black box with specific inputs and outputs (i.e. Input: Car geometry, flow conditions – Output: Flow field, drag force)

• To optimise an output (Objective Function) with respect to an input (Design Variables) we need the gradient

NUMERICAL OPTIMISATION

• An accurate and fast way to compute the gradient is needed!!

• A way is with Finite Differences: 𝑑𝑑𝑓𝑓𝑑𝑑𝑥𝑥 =

𝑓𝑓+ − 𝑓𝑓−𝑥𝑥+ − 𝑥𝑥−

5 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

ADJOINT METHOD

• Primal Equations • Objective Function • Differentiation of these:

INTRODUCTION

6 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

ADJOINT METHOD

• Direct Problem

• Adjoint Problem

INTRODUCTION

7 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Navier-Stokes Equations:

• Generic objective Function

• Augmented objective functions

8 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Differentiation of the augmented function: • where:

9 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Product Rule:

• Green-Gauss theorem:

10 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Applying these rules:

11 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Applying these rules:

12 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

,

“Adjoint Methods for Turbulent Flows, Applied to Shape or Topology

Optimization and Robust Design”, E. Papoutsis-Kiachagias, PhD Thesis

13 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Field Adjoint Equations

14 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Boundary conditions at inlet:

15 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Boundary condition at outlet:

16 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Boundary condition at walls:

17 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

• Sensitivity derivatives:

18 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Navier-Stokes for topology optimisation:

• Extra “porosity” term:

TOPOLOGY OPTIMISATION

19 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

CONTINUOUS ADJOINT FORMULATION

• Field adjoint equations with porosity:

• Sensitivities wrt. porosity:

TOPOLOGY OPTIMISATION

20 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

APPLICATIONS

• Objective Function: Power Losses

• Field Adjoint Equations

PITZ DAILY

21 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

APPLICATIONS

• Inlet boundary condition:

• Outlet boundary condition:

• Sensitivity derivatives:

PITZ DAILY

22 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

APPLICATIONS

• Objective Function: Drag

• Field Adjoint Equations

2D AIRFOIL

23 Group Research| Vehicle & Electronics| Vehicle Technology / CAE Methods (K-GERFG/V) | Christos Kapellos

APPLICATIONS

• Wall boundary condition:

• Outlet boundary condition:

• Sensitivity derivatives:

2D AIRFOIL